Light Path node works weirdly for Background
Using a Mix Shader node and 2 background nodes will give you better results and more flexibility than multiplying the strength of the entire world. you can also sorta &quot;mask&quot; it out and add whatever youd like on the 2nd background node and show one thing in the sky but for diffuse light bounces to react to another thing.

but if you absolutely need to use the node's strength, then doing it without any sort of multiply works, if u need to multiply it, then add a multiply node and/or an invert node to change the background's strength, with shouldnt be a problem with my method since the strength socket is available
EDIT: with what u said in comments, that would need to be done in the compositing of the scene. you would need to enable diffuse layers for indirect diffuse, then use that as a mask/factor for the other ndoes



this will take all light bounces coming from diffuse surfaces to the camera and use them as a factor, take note this will also affect other diffuse indirect bounces though. or, you can play around with materials and use the light path node on them and experiment a little